A formula for $\sum^n_{i=1}(1+1/n)$?

Question

Find a formula for $$\sum^n_{i=1}\left(1 + \dfrac{1}{n}\right)$$ Prove that it holds for all $n \geq 1$. It kind of looks like is a series but I didn't succeed in this problem. Can someone help me please. Thanks.

Answer 1

Assuming that you effectively want to explicit $$S=\sum^n_{i=1}\left(1 + \dfrac{1}{n}\right)$$ just think about what you do when summing. You just write $$S=\sum^n_{i=1}\left(1 + \dfrac{1}{n}\right)=\left(1 + \dfrac{1}{n}\right)+\left(1 + \dfrac{1}{n}\right)+\left(1 + \dfrac{1}{n}\right)+...+\left(1 + \dfrac{1}{n}\right)$$ what you made $n$ times. So the result of the summation is $S=\left(1 + \dfrac{1}{n}\right) \times n=(n+1)$.

Since you did not clarify the question, we could also have considered $$S=\sum^n_{i=1}\left(1 + \dfrac{1}{i}\right)$$ which is a very different story. For this case, expanding we have $$S=\sum^n_{i=1} 1 +\sum^n_{i=1}\dfrac{1}{i}=n+H_n$$ where appears the harmonic number.